本项目拟研究双曲矩阵分解的理论、计算、扰动分析及应用等。首先，根据经典内积与不定内积空间中矩阵的相关理论、(J_1,J_2)-正交阵和(J_1,J_2)-酉阵的性质、现有双曲矩阵分解的相关结论以及相关经典矩阵分解的特点等构建新型的双曲矩阵分解，并探讨其性质、存在性与唯一性条件等；其次，借助双曲Householder变换、双曲Givens旋转、牛顿迭代、Jacobi迭代以及它们的变型等探索新构建的双曲矩阵分解的计算方法，构造高效的算法；再次，针对范数型和分量型扰动，利用经典的矩阵方程方法、精致的矩阵方程方法、矩阵向量方程方法以及它们的组合等研究现有与新构建的双曲矩阵分解的扰动问题，以期获得能真切反映扰动影响的一阶与严格的绝对扰动界、相对扰动界、乘法扰动界及条件数估计等；最后，探讨部分双曲矩阵分解在不定最小二乘问题、约束不定最小二乘问题、约束不定二次规划问题、广义不定线性模型问题等上的应用。

This project will study the theory, computations, perturbation analysis, and applications of hyperbolic matrix factorizations. Firstly, we will propose some new hyperbolic matrix factorizations according to the related theory of matrices in the classical and indefinite inner product spaces, the properties of (J_1,J_2)-orthogonal and (J_1,J_2)-unitary matrices, the related results of the existing hyperbolic matrix factorizations, and the characteristics of the related classical matrix factorizations, and then discuss their properties and existence and uniqueness conditions. Secondly, we will explore the computation methods of the new proposed hyperbolic matrix factorizations using the hyperbolic Householder transformation, the hyperbolic Givens rotation, Newton iteration, Jacobi iteration, and their variants, and then construct efficient algorithms. Thirdly, for the norm-wise and component-wise perturbations, we will consider the perturbation analysis of the existing and new proposed hyperbolic matrix factorizations using the classical matrix equation approach, the refined matrix equation approach, the matrix vector equation approach, and their combinations, and then derive the first order and rigorous absolute perturbation bounds, relative perturbation bounds, multiplicative perturbation bounds, and condition number estimators which can accurately reflect the influence of perturbation. Finally, we will discuss the applications of some new hyperbolic matrix factorizations in some problems such as the indefinite least squares problem, the indefinite least squares problem with constraints, the indefinite quadratic optimization problem with constraints, and the generalized indefinite linear model problem.